On the Tightness of the Laplace Approximation for Statistical Inference

TL;DR

This paper investigates the accuracy of Laplace's method in statistical inference, establishing that its approximation error rate is optimally tight at $O_p(n^{-1})$, supported by new lower bounds.

Contribution

The paper provides the first statistical lower bounds demonstrating that the $O_p(n^{-1})$ error rate of Laplace's approximation is tight and cannot be improved.

Findings

Laplace's method has a relative error rate of $O_p(n^{-1})$

The $n^{-1}$ rate is proven to be the best possible (tight)

First statistical lower bounds confirming the tightness of the approximation error

Abstract

Laplace's method is used to approximate intractable integrals in a statistical problems. The relative error rate of the approximation is not worse than $O_p(n^{-1})$. We provide the first statistical lower bounds showing that the $n^{-1}$ rate is tight.